Question: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{r^2 - 9}{r - 3}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{9} = -3$ So we can rewrite the expression as: $t = \dfrac{({r} {-3})({r} + {3})} {r - 3} $ We can divide the numerator and denominator by $(r - 3)$ on condition that $r \neq 3$ Therefore $t = r + 3; r \neq 3$